textbook and course materials for 21-127 concepts of mathematics

21-127 History Content Comparison End

Textbook and course materials for 21-127Concepts of Mathematics

Thesis defense, D.A. in Mathematical Sciences

Brendan W. Sullivan

Carnegie Mellon University

May 8, 2013

Brendan W. Sullivan Carnegie Mellon UniversityTextbook and course materials for 21-127 Concepts of Mathematics


Abstract

Concepts of Mathematics (21-127 at CMU) is a course designed tointroduce students to the world of abstract mathematics, guiding themfrom more calculation-based math (that one might learn in highschool) to higher mathematics, which focuses more on abstractthinking, problem solving, and writing proofs. This transition tendsto be a shock: new notation, terminology, and expectations(particularly the requirement of writing mathematical thoughtsformally) give students much trouble. Many standard texts for thiscourse are found to be too dense, dry and formal by the students.Ultimately, this means these texts are unhelpful for student learning,despite their intentions. For this project, I have written a newtextbook designed specifically for this course and the way it has beentaught. My emphases—writing more informal prose, properlymotivating all new material, and including a wider variety of exercisesand questions for the reader—all reflect the goal of increasing thereader’s interest and potential for learning. I have also includedaccompanying class notes, meant to supplement the teaching of thiscourse, as well as assignments and exams, and their correspondingsolutions and grading rubrics.



Acknowledgements

This is joint work with Profs. John Mackey and Jack Schaeffer.

Thanks to John, Jack, Bill, and Hilary for guidance.

Thanks to all of you for attending.



1 Concepts of Math 21-127Course GoalsContent

2 Project HistoryTimelineRetrospective

3 Outline of Project ContentChaptersStyleExercises

Supplements4 Comparison to ExtantTexts

PositivesNegativesNovelties

5 Summary & ConclusionsEnd ResultsFuture WorkReferences



Course description

21-127 Concepts of MathematicsAll Semesters: 10 unitsThis course introduces the basic concepts, ideas and toolsinvolved in doing mathematics. As such, its main focus is onpresenting informal logic, and the methods of mathematicalproof. These subjects are closely related to the application ofmathematics in many areas, particularly computer science.Topics discussed include a basic introduction to elementarynumber theory, induction, the algebra of sets, relations,equivalence relations, congruences, partitions, and functions,including injections, surjections, and bijections. A prerequisitefor 15-211. (3 hrs. lec., 2 hrs. rec.)http://coursecatalog.web.cmu.edu/melloncollegeofscience/departmentofmathematicalsciences/courses/


http://coursecatalog.web.cmu.edu/melloncollegeofscience/departmentofmathematicalsciences/courses/

http://coursecatalog.web.cmu.edu/melloncollegeofscience/departmentofmathematicalsciences/courses/


Course Goals

Transition from computation to proof

High school math courses emphasize formulaic, roteapplication of methods: “Follow this example”.Even college calculus is focused on getting the right answerby standard techniques: “Plug and chug”.This is not what mathematics is truly about!We want students to be able to think abstractly,understand and generalize methods, explore variousproblem-solving techniques, and ultimately write proofs.This is a difficult transition for the students, and places aburden on the instructor and TAs.



Course Goals

Who takes this course, and why?

All years, but mostly freshmen/sophomores.(Have had grads and advanced high-schoolers, too.)All majors, but mostly MCS and CIT.Pre-req for most higher math courses.Required for math majors.Co-req for ECE 18-240.Pre-req for CS 15-150, co-req for 15-122.AP/EA students who want to break the “calculus cycle”.From our perspective: We want students to understandwhat math really is and what mathematicians do.Reputation: Challenging but rewarding course. Somestudents take it several times.



Content

Material covered

“Topics discussed include a basic introduction to elementary number theory,induction, the algebra of sets, relations, equivalence relations, congruences,partitions, and functions, including injections, surjections, and bijections.”(“The easier it sounds, the harder it is.”)

This allows for some variability, by instructor.Main goal is to introduce proof methods and illustratethem by exploring different topics.Certain fundamental topics need to be introduced here sothat students are familiar with them later, e.g. induction,sets, functions.Other topics can be chosen and developed based on interestor timing, e.g. cardinality, number theory, probability.



Content

Sequencing of material

(1) Mathematical arguments(2) Induction(3) Sets(4) Formal logic(5) Proof techniques(6) Relations(7) Equivalence classes

(8) Modular arithmetic(9) Number theory(10) Functions(11) Cardinality(12) Combinatorics(13) Probability(14) Basic graph theory

These can be reordered and/or dropped in some ways, but thetopics in the left column should precede those on the right.

Timing of the semester and pace of the course affects the levelof detail of included topics, as well.



Content

Organization and assessment

Three 1-hr lectures, MWFLed by instructor, introducesnew material and examples.

Two 1-hr recitations, TRLed by TAs, work onproblem-solving andconceptual help.

10 written assignmentsEmphasis on solvingproblems and writing proofs.

3 in-class examsEmphasis on content recalland writing proof detailsquickly (“time crunch”).

1 finalEmphasis on synthesizingcourse content and applyingmethods to write proofsacross areas.



Genesis and motivation

Served as TA for 21-127. Found myself agreeing with somestudents’ complaints about the standard text. Starteddeveloping ways to work around these conditions to presentbetter material and help student learning.Was looking for a project to devote myself to. Realized Iwas very interested in teaching and already felt motivatedto spend lots of time on course design and materials.Recognized how important this course is to the mathcurriculum, in general. Wished I had such a course in myundergrad education.



Timeline

Teaching

Fall 2010 — TASummer I 2011 — graderSummer II 2011 — instructorFall 2011 — TA (w/ Mackey)Summer 2012 — instructorFall 2012 — instructor (w/ Schaeffer)

While not teaching, writing!



Timeline

Writing

Spring 2011 — Began writing with Mackey. First three chapterstook shape. Stated goals and developed rough plan for the rest.Editing and guidance from Mackey based on his experience.

Summer 2011 — Used new materials to supplement standardtext while instructing. Focused on finding good motivatingexamples for material, as well as writing exercises to “field test”.

Fall 2011 — Spent much time helping with logistics of course,including grading (and verifications) and student help. Wrote“recitation sheets” whose content has been repurposed. Also usedsome of my written problems on homeworks.

Spring 2012 — Wrote most of the later chapters. Still decidingwhat to include/omit and how to sequence the material.



Timeline

Writing

Summer 2012 — Used solely my materials while instructing.Large summer class with many AP/EA students. Realized thatgreater emphasis on proof techniques is needed. Wrote lecturenotes and developed many examples and exercises to be used.

Fall 2012 — Used my materials and collaborated with Prof.Schaeffer on assignments/exams. Further developed lecture andrecitation notes, and exercises, which were very useful in finishingthe writing of the book and providing supplemental materials.

Spring 2013 — Took teaching experiences and written materialsand finished writing book. Decided on sequencing, examples andexercises. Standardized formatting and chapter structures,created many diagrams.



Retrospective

Course/book tension

Much of this project developed around my simultaneousteaching of the course, in various settings.Good:

Ability to “field test” examples, explanations and exercises.Instant feedback from students on materials, andgiven-and-take from me (via AnnotateMyPDF).

Bad:Had hopes for extra material that was not approached dueto time/space constraints.Might limit appeal of the book to outside sources, or evenother instructors.



Retrospective

Textbook/informality tension

As described later, one major goal is to write less formally thanother texts to engage the reader.Good:

This idea developed from direct experiences with students,and it seemed like this is what they wanted (or wouldunknowingly benefit from).Makes the book stand out amongst others; non-standardfor a math book designed for budding math-doers.

Bad:Might “turn off” especially rigorous students who werealready motivated to pursue higher mathematics.Length and “wordiness” can be overwhelming.



Overall structure of book

Part I: Learning to Think MathematicallyIntroduction to mathematical arguments (What is a proof?)including inductive argumentsSets as fundamental objects, new definitions and notationLogic and proof techniques (central focus)Return to induction, formalization and further development

Part II: Learning Mathematical TopicsApplying proof techniques to other topics while usingproblem-solving skills and gathering knowledgeRelations and properties, equivalence relations and classes,modular arithmetic, number theoryFunctions and properties, cardinality, infinite setsCombinatorics, basic counting arguments, counting in twoways, intro to advanced topics

(See other handout, TOC)Brendan W. Sullivan Carnegie Mellon UniversityTextbook and course materials for 21-127 Concepts of Mathematics


Chapters

Part I

1. What is Mathematics?2. Mathematical Induction: “And so on . . . ”3. Sets: Mathematical Foundations4. Logic: The Mathematical Language5. Rigorous Mathematical Induction: A Formal Restatement



Chapters

Part II

6. Relations and Modular Arithmetic: Structuring Sets andProving Facts About The Integers

7. Functions and Cardinality: Inputs, Outputs, and The Sizesof Sets

8. Combinatorics: Counting Stuff



Chapters

1. What is Mathematics?

1.1 Present four “proofs” of the Pythagorean theorem.Discuss prime numbers and the infinitude of primes.Argue there exist irrational a, b such that ab is rational.

1.2 Discuss importance of notation and definitions.Informally discuss logic and show bad/incorrect proofs.Dicuss some history and preview chapter on logic.

1.3 Describe presumed knowledge: familiarity with arithmeticand high-school level algebra (showing practice examples).Solve systems of equations, derive Quadratic Formula.Discuss standard sets of numbers, preview chapter on sets.



Chapters

1. What is Mathematics?

1.4 Present four interesting puzzles. Thoroughly discusssolutions, as well as how to approach them. Summarize eachwith a lesson.

The Missing Dollar ProblemGauss’ Sum: 1 + 2 + 3 + · · ·+ n = n(n+1)

2

Sum of odds: 1 + 3 + 5 + · · ·+ (2n− 1) = n2

Monty Hall Problem

1.5 Exercises: Algebraic problems, simple puzzles that requiresome ingenuity, practice with definitions and notation.

1.6 Lookahead: Focus on the sum puzzles, inductive arguments.



Chapters

2. Mathematical Induction

2.1 Intro: objectives, segue, motivation, goals and warnings.2.2 Examples of “regular induction”:

∑k2 and lines on a plane

2.3 Domino Analogy (and others), an informal discussion2.4 Examples of “strong induction”: Fibonacci tilings and

“Takeaway” (basic Nim, handout 1, p. 131)2.5 Applications: recursive programming, Tower of Hanoi2.6 Summary2.7 Exercises: inductive arguments, guiding through induction

proofs, discovery of formulae, several “spoofs”2.8 Lookahead: develop the fundamentals to be more rigorous



Chapters

3. Sets

3.1 Intro: objectives, segue, motivation, goals and warnings.3.2 Main idea: examples from everyday life and math3.3 Definition and examples: proper notation, set-builder usage,

empty set, Russell’s Paradox (and a brief note on axioms)3.4 Subsets: definition and standard examples, finding the

power set, equality by containment, the “bag analogy”3.5 Set operations: ∩,∪,−, A (handout 4, p. 174)3.6 Indexed sets: notation, usage, examples, operations3.7 Cartesian products: ordered pairs, examples3.8 [Optional] Defining N via sets: inductive sets, state PMI



Chapters

3. Sets

3.9 Proofs involving sets:Introduces idea of appealing to formal definitions.How to prove ⊆Double-containment proofs to show =Many examples, including indexed operationsDisproving claims (introduces logical negation gently)

3.10 Summary3.11 Exercises: practice with notation and reading statements,

asks reader to provide examples and non-examples, several“spoofs”, proofs involving sets (avoiding arguments thatwould be made easier via logic, e.g. DeMorgan’s Laws)

3.12 Lookahead: logical ideas, develop proof techniques to apply



Chapters

4. Logic

4.1 Intro: objectives, segue, motivation, goals and warnings.4.2 Mathematical statements: variable propositions, examples

and non-examples, proper notation4.3 Quantifiers: usage and notation, how to read and write

statements, “fixed” variables4.4 Negating quantifiers: method and examples, Law of the

Excluded Middle, redefine indexed set operations4.5 Connectives: ∧,∨, =⇒ ,⇐⇒

Many examples and non-examples of each, in-depthdiscussion of =⇒ and various forms, redefine set operations



Chapters

4. Logic

4.6 Logical equivalence: definition and usage and examples,biconditionals, necessary/sufficient, associative/distributivelaws, DeMorgan’s Laws, (double-)containment proofs

4.7 Logical negations: use DeMorgan’s Laws, negatingP =⇒ Q, method and examples

4.8 [Optional] Truth sets: relating connectives to sets4.9 Proof strategies: major focus, outline direct/indirect/other

methods for each connective, implement an exampleshowing necessary scratch work (handout 3, p. 286)

4.10 Summary4.11 Exercises: applying proof techniques, discovering truths4.12 Lookahead: have learned ideas to revisit induction



Chapters

5. Formal Mathematical Induction

5.1 Intro: objectives.5.2 Regular induction: state and prove PMI, provide proof

template and illustrate usage, emphasize common errors5.3 Variants: different base case, backwards, evens/odds5.4 Strong induction: state and prove PSMI, provide proof

template and illustrate usage, compare to PMI5.5 Variants: “minimal criminal”, WOP, TFAE5.6 Summary5.7 Exercises: more difficult arguments, prove WOP, “spoofs”5.8 Lookahead: new goal, functions

This concludes Part I (just over halfway).



Chapters

6. Relations and Modular Arithmetic

6.1 Intro: objectives, segue, motivation, goals and warnings.6.2 Binary relations: definition and examples, properties

(reflexive, symmetric, transitive, anti-symmetric) andcanonical examples, proving/disproving properties

6.3 [Optional] Order relations: posets, tosets, chains, (toinclude: well orders)

6.4 Equivalence relations: examples and motivation, equivalenceclasses and how to characterize them, partitions, theorems(some proofs as exercises)



Chapters

6. Relations and Modular Arithmetic

6.5 Modular arithmetic: equivalence classes mod n, using modsto prove facts, multiplicative inverses (relatively prime),solving Diophantine equations, CRT, Bézout

6.6 Summary6.7 Exercises: proving properties of relations, characterizing

equivalence classes, solving number theory claims, provinginteresting lemmas and theorems, some “spoofs”

6.8 Lookahead: a function is a relation



Chapters

7. Functions and Cardinality

7.1 Intro: objectives, segue, motivation, goals and warnings.7.2 Definition and examples: “well-defined”, functional equality,

schematics, breaking idea that it’s a “rule” on numbers7.3 Images and pre-images: definitions and easy/hard examples,

proof strategies (reiterate sets and logic), constructingcounterexamples (handout 2, p. 488)

7.4 Properties: “jections”, definitions and examples, proofstrategies, how to determine properties

7.5 Compositions and inverses: notation and usage andexamples, proving inverse by composing both ways to getidentity, bijection iff invertible



Chapters

7. Functions and Cardinality

7.6 Cardinality: finite vs. infinite, comparing via functions(discussion on axioms/defns), Cantor’s Theorem, countablyinfinite sets (Hilbert Hotel, examples and theorems, infinitevs. arbitrarily large), uncountably infinite sets (examplesand theorems, levels of infinity)

7.7 Summary7.8 Exercises: wide range of difficulties, many lemmas from

chapter, exploring properties, prove/find counterexample,“spoofs”, sets of binary strings (handout 5, p. 556)

7.9 Lookahead: focus on finite sets



Chapters

8. Combinatorics

8.1 Intro: objectives, segue, motivation, goals and warnings.8.2 Basic counting principles: Rules of Sum and Product,

fundamental objects and formulae (permutations, selections,binomial coefficients, arrangements), relation to functions

8.3 Counting arguments: combining ROS/ROP, case analysis,decision processes, being careful of under/overcounts, otherobjects (n-tuples, alphabets, anagrams, lattice paths)

8.4 Counting in two ways: method summary, examples,theorems and uses, how to analyze an identity and constructan argument

8.5 Selections with repetition: Pirates & Gold (stars & bars),balls in bins, indistinguishable dice, examples



Chapters

8. Combinatorics

8.6 Pigeonhole: statement and proof (logic), examples8.7 Inclusion/Exclusion: statement and proof, examples where

intersections have fixed size, other examples (sels. w/rep.)8.8 Summary8.9 Exercises: wide variety of difficulties, many counting in two

ways, some “spoofs” to identiy over/undercounts, Fermat’sLittle via binomial coeffs, comparing selections with andw/o repetition

8.10 Lookahead: go forth and prosper



Chapters

Appendix

Definitions and Theorems

Separated by topic (not necessarily chronology)

Proof strategies for connectives, functions, induction

Cardinality catalog

Acronyms and phrases

Helpful reference (suggested by students)



Style

Prose: readable, engaging, but meaningful

See handout 1, p. 131-134.Encourage reader to explore the concepts and examples ontheir own. Ask insightful questions to guide them.Present motivation and analysis as if in the role of aknowledgeable fellow student, but backed by expert insight.Admittedly, writing looks “texty” but this is broken up bysection headings, itemized lists, diagrams, and questions.Encouraging the reader: we are on the same journey.Written as if I were speaking in the classroom, but benefitsfrom organization and foresight.Not an informational reference necessarily, but a conceptualreference. (Where else can students find fully explained andmotivated theorems and examples?)



Style

Information: avoiding “expert blind-spots”

See handout 2, p. 488-491.Mathematical writing tends to treat the reader as a fellowexpert, with no mention of work “behind the scenes”.“We often skip or combine critical steps when we teach. Students, on the

other hand, don’t yet have sufficient background and experience to make

these leaps and can become confused, draw incorrect conclusions, or fail to

develop important skills. They need instructors to break tasks into

component steps, explain connections explicitly, and model processes in

detail.” Eberly Center: http://www.cmu.edu/teaching/principles/teaching.html

Somewhat like the “follow this example” method butemphasizes critical thinking, not a rote algorithm.See handout 3, p. 286-291 for a combo of these approaches.


http://www.cmu.edu/teaching/principles/teaching.html


Style

Consistency of chapter structures

Each chapter has the following outline:Introduction:

ObjectivesSegue from previous chapterMotivationGoals and warnings for the reader

Sections:ContentQuestions: Remind YourselfExercises: Try It

Conclusion:SummaryChapter exercisesLookahead



Style

Consistency of chapter structures

Example of objectives (from Chapter 8: Combinatorics):By the end of this chapter, you should be able to . . .

State the Rules of Sum and Product, and use and combine them toconstruct simple counting arguments.Categorize several standard counting objects, as well as state correspondingcounting formulas and understand how to prove them.Understand the meaning of binomial coefficients, how to use them incounting arguments, and how to derive their numerical formula.Critique a proposed counting argument by properly demonstrating if it is anundercount or overcount.Prove combinatorial identities by constructing “counting in two ways” proofs.Understand various formulations of selection with repetition, and use themto solve problems.State the Pigeonhole Principle and use it in counting arguments.State the Principle of Inclusion/Exclusion and use it in counting arguments.



Exercises

Section exercises: easy concept checks

See handout 4, p. 174-175Questions: Remind YourselfChecking the ability to recall a definition or theorem, useproper notation, identify the difference between twoconcepts, name canonical examples/non-examples.Exercises: Try ItRequire some more thought/effort but are not meant to betoo challenging. Reminds the reader they need to do math.Together, summarizing and reinforcing main ideas from thesection. Building up understanding to move on with morecontent. Easing into more difficult chapter exercises.



Exercises

Chapter exercises: variety and synthesis

See handout 5, p. 556-565.Combines material from that chapter and all previous.Problems range widely in difficulty. (Typically, easiest onescome first; remainder are spread out.)Easy problems can amount to understanding definitionsand notation.Always features a few “spoofs”: find the flaw (if any!) in aproposed argument. Essential mathematical skill.Often asks reader to prove lemmas/theorems we stated.Several prove/disprove problems.Some difficult problems scaffolded to guide the reader (e.g.7.8.30, p. 561, structured double induction).Various hints and suggestions.



Supplements

Class notes

Lecture notes:Condensed versions of definitions and examples, with somediscussion. Some new examples and questions.Helpful for instructor to set pace/timing of course.Helpful for student to have (perhaps) more digestible notes.Recitation notes:Extra examples and exercises to work through.Supplements course material with problems that wouldn’tsqueeze into a lecture (time/content).(Two versions: one for TA notes, one for student handout.)Helpful for instructor/TAs to have suggested problems.Helpful for student to have written record of (often)predominantly verbal problem-solving sessions.



Supplements

Homeworks and exams

Most homework problems appear in book.Helpful for instructor to have identified set of problems thatare especially fruitful for students to work on.Most exam problems do not appear in book.Helpful for instructor to see what can feasibly be completedin 50 minutes. Exams carefully designed to minimize timecrunch (Prep Questions) and address all required skills(concepts, read math, problem-solve, write proofs).Solutions and rubrics included.Helpful for student to have fully detailed examples of goodproofs. Need role models.Helpful for instructor/TAs to have indication of what isimportant in a solution (based on common errors).



Why bother?

Aren’t there many books/courses that do this?

I knew there could be a better book, one that would be morehelpful for students and would bring more of them into the worldof mathematics. (In particular, I wish I’d had such a book.)

“The D.A. thesis . . . [is expected] to demonstrate an ability to organize, understand, and

present mathematical ideas in a scholarly way, usually with sufficient originality and worth to

produce publishable work.”

http://www.math.cmu.edu/graduate/PhDprogram.html


http://www.math.cmu.edu/graduate/PhDprogram.html


Positives

Addresses teaching principles

Effective teaching involves . . .http://www.cmu.edu/teaching/principles/teaching.html

. . . acquiring relevant knowledge about students and usingthat knowledge to inform our course design and classroomteaching.. . . articulating explicit expectations regarding learningobjectives and policies.. . . prioritizing the knowledge and skills we choose to focuson. (“Coverage is the enemy.”)

. . . recognizing and overcoming our expert blind spots.

. . . adopting appropriate teaching roles to support ourlearning goals.


http://www.cmu.edu/teaching/principles/teaching.html


Positives

Made from experience and trial

Much of the content was developed first in recitations,seeing where students needed more/fewer examples,more/less illustration of theorems/proofs vs. applications,and gathering helpful heuristic explanations.This was developed into full-fledged lecture notes andtextbook writing.Likewise, interactions in office hours, plus lots of rubricwriting and grading, have informed decisions aboutexercises and expectations.Use of AnnotateMyPDF has brought suggestions andcriticisms directly from a variety of readers.“This example was helpful.” “I didn’t understand this theorem until . . . ”



Positives

Stands out as different

Casually flipping through, you can tell this is not a“standard” math text.Investigating further, you’ll (hopefully) find yourselfwanting to read more.(If you find it redundant/unnecessary, it wasn’t designedfor you in the first place, and would be better as a guide.)Math seems to make outsiders feel unworthy, or at leastmightily tests their mettle. I place my writing on the levelof a novice reader while simultaneously guiding them in.Books with similar prose style are aimed at the “amateurs”and “puzzlers”, not necessarily looking to bring them intothe world of abstract math.



Positives

Potential for both self-study and use in teaching

Engaging writing style could be perfect for a motivatedreader, even without the structure of a course/instructor.No knowledge presumed beyond familiarity with numbersand algebra, so could easily be recommended to adeveloped high-schooler wondering where to go.Varied difficulty in exercises can appropriately test anyonefrom a casual reader to a devoted student.Breadth of examples and exercises gives instructor lots ofchoices. Course can follow book closely or use it as areference for further development of material.Points out “lack of time/space” wherever relevant.Instructor could supplement these sections withnotes/problems.



Negatives

Potential issues with both self-study and use in teaching

Might be difficult to significantly alter sequencing of coursematerial, e.g. formal logic entirely before sets.Might be difficult to significantly alter pacing of coursematerial, e.g. length of chapters does not correlate toclassroom time, and e.g. whether or not to assign readings.Worry that a reader will expect all texts to be like this.Worry that engaging style might cut down on externalstudy and dedicated work (despite insistence otherwise).



Negatives

Readability vs. reference

“Wordy” writing might negatively affect future use as areference. Potential difficulties in locating material, wadingthrough no-longer-needed explanations.Many important aspects of the course are conceptual innature. Easy to look up definitions/theorems, but whatabout ideas and strategies?Will students who don’t necessarily need the extraexplanations then not bother to read, and be worse off?



Negatives

Informality can be confusing

Potential issues with ESL students: colloquialisms, culturalreferences, sheer volume of text can be off-putting.Am I just “passing the buck” on the transitional shock tohigher mathematics?Might this style dissuade students who were looking forcold, hard rigor from pursuing more math?



Novelties

“Sufficient originality and worth”

Detailed learning objectives and motivation within everychapter/section.Vast array of exercises to engage both a struggling noviceand a partial expert.Emphasis on reading in a book (how novel!).Written not only with the novice student as the intendedaudience but also ACTFTPOVAIFP.Content founded on rigorous mathematics and curriculargoals, while presentation founded on education research.Ultimately, believe the positives outweight the negatives,and they can both be massaged by attentive instruction.



End Results

What do we have now?

Textbook designed specifically (and “play-tested”) for use inthis course. Content sequencing, motivations, examples,and exercises designed from experience and usage.Supplemental materials to aid in the teaching and logisticalimplementation of this course.Book could serve as standalone text for motivated reader.Set of materials could serve as reference/inspiration for(future) instructors/TAs.



Future Work

Other ideas to explore

Further chapters:Probability. Graphs/discrete structures. More advancedcombinatorics. Intro to abstract algebra and groups. Introto analysis. Point-set topology. (“Concepts II”)Further content:Appendix with hints. Supplemental instructor’s solutions.More exercises (particularly challenging ones).Applying this writing style to other courses/material:More apt for “lower level” math, particularly courses meantfor non-majors. Could conceivably be useful on smallerscales in upper-level math, though.At least, rethink learning objectives and presentation style.Online access.



References

References

A. AaboeEpisodes From the Early History of MathematicsMAA, 1964

T. Andreescu, Z. FengA Path to Combinatorics for Undergraduates: Counting StrategiesBirkhäuser, 2004

J.P. D’Angelo, D.B. WestMathematical Thinking: Problem-Solving and Proofs.Prentice Hall, 2000

R.A. BrualdiIntroductory Combinatorics, 5th Ed.Prentice-Hall, 2010

C.M. CampbellIntroduction to Advanced Mathematics: A Guide to Understanding ProofsBrooks/Cole, 2012

C. Chuan-Chong, K. Khee-MengPrinciples and Techniques in CombinatoricsWorld Scientific, 2004



References

References

M.J. CullinaneA Transition to Mathematics with ProofsJones & Bartlett, 2013

A. CupillariThe Nuts and Bolts of Proofs: An Introduction to Mathematical ProofsElsevier, 2013

U. Daepp, P. GorkinReading, Writing, and Proving: A Closer Look at Mathematics, 2nd Ed.Springer, 2011

M. DayAn Introduction to Proofs and the Mathematical Vernacularhttp://www.math.vt.edu/people/day/ProofsBook/IPaMV.pdf

M. EricksonPearls of Discrete MathematicsCRC Press, 2010

S.S. EppDiscrete Mathematics with Applications, 3rd Ed.Brooks-Cole, 2004



References

References

A. GardinerDiscovering Mathematics: The Art of InvestigationClarendon Press, 1994

M. GardnerMathematical Puzzles of Sam LloydDover, 1959

M. GemiganiFinite ProbabilityAddison-Wesley, 1970

W.J. Gilbert, S.A. VanstoneClassical Algebra, 4th Ed.University of Waterloo, 2000

W.J. Gilbert, S.A. VanstoneAn Introduction to Mathematical Thinking: Algebra and Number SystemsPearson, 2005

A.M. GleasonFundamentals of Abstract AnalysisAddison-Wesley, 1966



References

References

J.E. HafstromIntroduction to Analysis and Abstract AlgebraW.B. Saunders Company, 1967

P.R. HalmosI Want to Be a MathematicianSpringer-Verlag, 1985

R. HammackBook of Proofhttp://www.people.vcu.edu/ rhammack/BookOfProof/

S.G. KrantzHow to Teach Mathematics, 2nd Ed.AMS, 1999

L.R. LieberThe Education of T.C. Mits (The Celebrated Man In The Street)W.W. Norton & Company, 1942

E. MenedelsonNumber Systems and the Foundations of AnalysisDover, 2001



References

References

D. NiedermanThe Puzzler’s DilemmaPerigree, 2012

C.S. Ogilvy, J.T. AndersonExcursions in Number TheoryDover, 1988

G. PolyaHow To Solve ItDoubleday, 1957

A.S. Posamentier, C.T. SalkindChallenging Problems in AlgebraDover, 1988

R.H. RedfieldAbstract Algebra: A Concrete IntroductionAddison-Wesley, 2001

D. SolowHow to Read and Do Proofs, 5th Ed.John Wiley & Sons, 2010



References

References

I. StewartConcepts of Modern MathematicsDover, 1995

R.R. StollSet Theory and LogicDover, 1961

A. TuckerApplied Combinatorics, 4th Ed.John Wiley & Sons, 2002

D.J. VellemanHow To Prove It: A Structured ApproachCambridge University Press, 1994

M.E. Watkins, J.L. MeyerPassage to Abstract MathematicsAddison-Wesley, 2012

P. WinklerMathematical Puzzles: A Connoisseur’s CollectionAK Peters, 2004



References

References

A.M. Yaglom, I.M. YaglomChallenging Mathematical Problems with Elementary Solutions, Volume 1:Combinatorial Analysis and Probability TheoryDover, 1964

Art of Problem SolvingArt of Problem Solvinghttp://www.artofproblemsolving.com/

J.L. Borges“The Library of Babel.” Collected Fictions (Trans. Andrew Hurley)Penguin, 1998

Cut The KnotInteractive Mathematics Miscellany and Puzzleshttp://www.cut-the-knot.org/

K. DevlinDevlin’s Angle: What is Mathematical Thinking?http://devlinsangle.blogspot.com/2012/08/what-is-mathematical-thinking.html



References

References

Eberly Center for Teaching Excellence & Educational InnovationThe Educational Value of Course-level Learning Objectives/Outcomeshttp://www.cmu.edu/teaching/resources/Teaching/CourseDesign/Objectives/CourseLearningObjectivesValue.pdf

Eberly Center for Teaching Excellence & Educational InnovationTeaching Principleshttp://www.cmu.edu/teaching/principles/teaching.html

S.S. EppThe Role of Logic in Teaching ProofAmerican Mathematical Monthly, 110 (2003) 886-899

P.R. HalmosWhat is Teaching?American Mathematical Monthly, 101 (1994) 848-854

P.R. HalmosThe problem of learning to teach American Mathematical Monthly, 82 (1975), 466Ð476

P.R. HalmosHow to write mathematicsEnseign. Math. (2), 16 (1970), 123Ð152.



References

References

P.R. HalmosMathematics as a creative artAmerican Scientist 56, (1968) 375Ð389

D. LittleOn Writing Proofshttp://www.math.dartmouth.edu/archive/m38s04/public_html/proof_writing.pdf

P. LockhartA Mathematician’s Lamenthttp://www.maa.org/devlin/LockhartsLament.pdf



References

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textbook and course materials for 21-127 concepts of mathematics

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